This paper studies Emerton's Jacquet module functor for locally analytic
representations of $p$-adic reductive groups, introduced in \cite{emerton-jacquet}.
When $P$ is a parabolic subgroup whose Levi factor $M$ is not commutative,
we show that passing to an isotypical subspace for the derived subgroup
of $M$ gives rise to essentially admissible locally analytic representations
of the torus $Z(M)$, which have a natural interpretation in terms of rigid
geometry. We use this to extend the construction in of eigenvarieties in
\cite{emerton-interpolation} by constructing eigenvarieties interpolating
automorphic representations whose local components at $p$ are not necessarily
principal series.